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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 1

### Complex $N$-Spin bordism of semifree circle actions and complex elliptic genera

Pages: 343 – 371

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a19

#### Author

#### Abstract

We give a complete bordism analysis of rational bordism groups of semifree circle actions on complex $N$-Spin manifolds. Moreover, we introduce the notion of a complex $N$-Spin$^{c,t}$ manifold and give a characterization of cobordism groups of such manifolds which we use to compute the rational bordism groups of free circle actions of type $t$ on complex $N$-Spin manifolds. Furthermore, we exploit this bordism analysis to furnish a mechanism with which we investigate a description, in terms of kernels of complex elliptic genera, of the ideal $I_*^{N,t}$, generated by bordism classes of connected complex $N$-Spin manifolds admitting an effective circle action of type $t$, in the rational complex $N$-Spin cobordism ring $\Omega_*^{U,N}\otimes\mathbb{Q}$.

#### Keywords

$U$-manifold, complex $N$-Spin manifold, complex elliptic genus of level $N$, the universal complex elliptic genus, circle action of type $t$, complex $N$-Spin$^{c,t}$ manifold, complex twisted projective bundle

#### 2010 Mathematics Subject Classification

Primary 53C27, 55N22, 57R77, 57R85, 58J26. Secondary 32Q60, 55N20, 55P62, 55R10, 57R20.